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Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems

Tudor C. Ionescu, Alessandro Astolfi

Abstract

In this paper we propose a solution to the problem of moment matching with preservation of the port Hamiltonian structure, in the framework of time-domain moment matching. We characterize several families of parameterized port Hamiltonian models that match the moments of a given port Hamiltonian system, at a set of finite interpolation points. We also discuss the problem of Markov parameters matching for linear systems as a moment matching problem for descriptor representations associated to the given system, at zero interpolation points. Solving this problem yields families of parameterized reduced order models that achieve Markov parameter matching. Finally, we apply these results to the port Hamiltonian case, resulting in families of parameterized reduced order port Hamiltonian approximations.

Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems

Abstract

In this paper we propose a solution to the problem of moment matching with preservation of the port Hamiltonian structure, in the framework of time-domain moment matching. We characterize several families of parameterized port Hamiltonian models that match the moments of a given port Hamiltonian system, at a set of finite interpolation points. We also discuss the problem of Markov parameters matching for linear systems as a moment matching problem for descriptor representations associated to the given system, at zero interpolation points. Solving this problem yields families of parameterized reduced order models that achieve Markov parameter matching. Finally, we apply these results to the port Hamiltonian case, resulting in families of parameterized reduced order port Hamiltonian approximations.

Paper Structure

This paper contains 17 sections, 22 theorems, 96 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Time-domain moment matching
  4. The MIMO case
  5. Krylov projections
  6. Markov parameter matching as moment matching for a class of descriptor systems
  7. Alternative approach
  8. Relation with Krylov methods
  9. Matching with preservation of the port Hamiltonian structure
  10. Matching at finite interpolation points
  11. Markov parameter matching with preservation of port Hamiltonian structure
  12. Illustrative examples
  13. SISO ladder network
  14. MIMO single machine infinite bus system
  15. Conclusions
  16. ...and 2 more sections

Key Result

Theorem 1

astolfi-TAC2010astolfi-CDC2010 The following statements hold.

Figures (8)

  • Figure 1: Block diagrams of systems (\ref{['exmp_markov_ABC']}) (a) and (\ref{['exmp_markov_ABC_1']}) (b).
  • Figure 2: Interconnection of the signal generator and the system (\ref{['sys_descript_2']}), with transfer function $\widetilde{K}(\tau)$.
  • Figure 3: Interconnection of the signal generator and the system (\ref{['sys_descript_2']}), with the transfer function $\widetilde{K}(\tau)$.
  • Figure 4: Interconnection of the signal generator and the system (\ref{['sys_descript_1']}), with the transfer function $\widetilde{K}(\tau)$.
  • Figure 5: Fourth order ladder network.
  • ...and 3 more figures

Theorems & Definitions (45)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Remark 1
  • Proposition 3
  • proof
  • Example 1
  • ...and 35 more